What is the relationship between UV image and material texture?
The material and its texture are what's used when you render the image. Anything else, like being able to see an image you have loaded into the Image-Editor in the 3D view are for when you want to preview your texture, maybe to see if the UVs are ok. 

It makes sense to be able to have a material set up with textures, but then also be able to view a single texture loaded from the Image-Editor to view the texture on its own. Otherwise, if you had a complex material with multiple textures, there would be no easy way to view a single texture on the model for inspection without removing some textures from the material.

What you show in your screenshots are 'Texture' shading mode:



Texture mode is for viewing textures, it loads in whatever image is in the Image-Editor.

If you change to 'Material' mode (shown in the list of viewport modes in that screenshot) you will see the material and any texture it may have (make sure you have a lamp to illuminate the object, otherwise it will appear black).

To summarise:


Texture mode is for seeing how an individual texture looks on your model, so you know if you have to make adjustments, if it doesn't look right.
The material and its textures will be used when you render your image (F12)
Materials are for the final look, 'Texture' viewport mode is for viewing a single texture.


As you can see from that screenshot, there are different viewport shading modes for different tasks. More information on these different modes and what they are suited to is in the Blender Manual, here.